I'm looking for an example of a ring $R$ (necessarily nonunital) which is simple (in the sense that $R \cdot R \neq 0$ and $R$ has no proper, nonzero 2-sided ideals) and also radical (in the sense that the Jacobson radical $J(R)$ is all of $R$). My only thought so far has been that it suffices to find a simple ring in which all elements are nilpotent. Thanks.